which is supposed to be the standard definition. But in the book The Geometry of Four-Manifolds (Donaldson-Kronheimer, page 76) one finds a rather different definition, at least for a 4-dimensional vector space ($S$ is supposed to be a general two-dimensional complex vector space with Hermitian metric and compatible complex symplectic form):

What is the meaning of the second definition? Everything seems quite involved and unrelated. Any idea will be helpful and welcome.

$\begingroup$The Donaldson-Kronheimer definition gathers in it all the facts about a spin structure that are needed in gauge theory. They all follow from the representation theory of Spin(4). In particular, they are needed to define the Dirac operator.$\endgroup$
– Liviu NicolaescuDec 6 '14 at 10:37

2 Answers
2

As it stands, the second definition is a concrete description of the spin group in dimension four. It defines an action of the simply connected group $SU(2)\times SU(2)$ on a vector space of real dimension four, which preserves a positive definite inner product, and this identifies $SU(2)\times SU(2)$ with the universal covering of the special orthogonal group of this four-dimensional Euclidean space.
To view it as a definition of a spin-structure on a manifold, one has to do this in each point of the manifold. This means that the spin strucuture in this sense is given by two auxiliary complex rank two bundles $S^+$ and $S^-$ which are endowed with Hermitian bundle metrics and compatible complex symplectic forms, togehter with an isomorphism between the tangent bundle and the bundle $Hom_J(S^+,S^-)$ which respects the inner products on the two spaces (the given Riemannian metric and the inner product constructed point-wise as in definition 2).
The equivalence between the two definitions is obtained as follows: To go from definition 1 to definition 2, you form the associated bundles corresponding to the two basic (complex) spin-representations of $Spin(n)$. In the opposite direction, you form the frame bundle of $S^+\oplus S^-$ (with structure group $SU(2)\times SU(2)$, which is isomorphic to $Spin(n)$) and the identification of $Hom_J(S^+,S^-)$ with $TM$ preserving bundle metrics shows that this bundle is a two fold covering of the $SO(n)$-frame bundle associated to the Riemannian metric.

$\begingroup$Which is the explicit definition of the action of $SU(2)\times SU(2)$ on $V$ given the isomorphism $\gamma:V\longrightarrow Hom_J(S^+,S^-)$?$\endgroup$
– JjmJan 16 '15 at 10:02

1

$\begingroup$You just take $V$ to be the space $Hom_J(S^+,S^-)$. This is a real vector space of dimension $4$ endowed with an inner product, and the action of $SU(2)\times SU(2)$ on that space by construction preserves that inner product. Therefore you get a homomorphism to $SO(V)$ and you have to verify that this is a two-fold covering.$\endgroup$
– Andreas CapJan 16 '15 at 11:11

$\begingroup$Take the first copy of $SU(2)$ as quaternionic automorphisms of $S^+$ and the second as quaternionic automorphisms of $S^-$. Then the action is given by composition from both sides, i.e. $((A,B)\cdot f)(x)=B(f(A^{-1}x))$.$\endgroup$
– Andreas CapJan 16 '15 at 11:35

As Liviu says, these properties follow from the usual definition of spin structures (in dimension 4). It's a little more work to prove that the existence of these bundles with Clifford multiplication gives a spin structure in the usual sense. Details are given in the Kronheimer-Mrowka book Monopoles and 3-manifolds. They do spin^c structures there, which are even more useful.

The discussion is a little easier in dimension 3 (op. cit.). I don't know if there is an analogous way of defining spin and spin^c structures in other dimensions.